Optimal Positioning of Pseudolites Augmented With GPS

September 6, 2008

By Saka, M Halis

Finding an Optimum Location for a Fixed Pseudolite In Support of CPS Navigation in Poor Visibility Conditions Over the last decade or so, global positioning system (GPS) positioning has played an escalating role in navigation, and it has become a primary tool for precise positioning. However, in some applications (e.g., urban canyons, valleys, deep open-cut mines and near-shore applications) the number of visible satellites may not be sufficient to reliably determine the user’s position. In addition, GPS satellite availability tends to be a function of the observer latitudes, and some techniques (e.g., phase smoothing) are sensitive to cycle slips, hence they require continuous data acquisition.1 Pseudo satellites, known as pseudolites (PLs), provide extra and continuous signals to GPS users so that they can significantly improve the satellite geometry and positional accuracy. PLs typically transmit pseudorange and carrier phase signals from the ground at Ll and L2 GPS frequency bands. Normally, standard GPS receivers with minor firmware modifications can track PL signals.

The placement of a PL with respect to the user’s location can be critical. Therefore, an optimization approach that quantifies the level of improvement with respect to the geometry of the GPS satellites together with PLs can provide considerable benefits.’ The philosophy behind the optimization approach is to obtain an optimal geometry, assuring the minimum in terms of a cost function. Additionally, the effect of the signal masking due to the local horizon or man-made structures is a constraint in the placement of PLs.

In this study, two optimization strategies are described and applied to a data set using different geometrical dilution of precision (GDOP) criteria. The fundamental idea behind these strategies is to observe the maximum GDOP and total GDOP variations, then to detect the critical directions producing optimal GPS-PL configurations. The results of these considerations are compared to verify their effectiveness, especially for the case where no PL was employed.

Background Information

A typical satellite-PL configuration was considered for the selected region in this study. Such a system is not reliable when the configuration of the satellites is weak due to the number of visible satellites and their geometrical configuration. In this case, one must support the system by adding a ground PL to the system. At this point, an important question must be asked: Which location is the optimal one for the PL?

The linearized pseudorange equation can be expressed as equation one (shown top of page); where ? is a vector of pseudorange observations, ? is the vector of residuals, G is the design matrix that explains the satellite geometry related to the user’s position, ? shows the user’s position and parameters of the receiver clock offset and p” is the approximate range between the user and the satellites. The design matrix, G, can be written using the azimuth and elevation of the satellites in the form of equation two.

Although various cost functions are suggested in the literature, the most popular one is the GDOP, which is used as a precision indicator in GPS applications.1 GDOP is estimated from the design matrix, C. The minimum of the GDOP corresponds to the best geometry of the observed satellites. Therefore, the optimization of new PL locations should minimize the GDOP (i.e., GDOP=min). The PL location with respect to the user is defined by its azimuth An) and elevation angles En), As a cost function (f), GDOP can be estimated as equation three.

Note that the parameters of En and An are the unknowns, which are the design variables for the optimization. The remaining coefficients of C are known constants for given satellite configurations. In order to minimize f(An, En), the necessary conditions are shown in equations four and five. Solving the nonlinear equations four and five yields the solution of An and En that minimize the cost function in equation three. It should be pointed out that the solutions obtained from equations four and five belong to an instant of time.

In practical applications, the solutions for a period of time (e.g., 24 hours) are required. In this case, the new cost function converts in the form of equation six, where i refers to each instant of time in the simulation data and nt shows the number of sampling data for 24 hours.

Obviously, obtaining the analytical solution using the cost functions is computationally expensive due to the complexity of calculating the inverse of the term G’(i)G(i) in equation six. In order to overcome this problem, efficient numerical approaches are required.

Numerical Optimization Methods

The GDOP differences between the optimal and pessimal positioning of the PL can help find the most sensitive geometry of satellite constellation with the inclusion of the PL. Maximum improvement in GDOP can be achieved by positioning the PL for the instant when the geometry is most sensitive. However, it should be pointed out that this does not guarantee the lowest GDOP for all epochs. The optimal and pessimal locations of the PL in terms of the GDOP criterion were estimated for the data set. It was found that the use of a PL can make significant improvements in terms of GDOP with a good selection of its location, even for the case where a maximum number of visible satellites was used. This is valid for the positioning of a fully mobile PL. However, in real-world applications, a fixed location is required for PLs. Sometimes, though rarely, the use of a PL may not provide any improvement when it is placed at the worst location, as in the data set for GPS elevations greater than 23[degrees].

In order to select the best location of a PL for a specified time span, robust approaches or strategies are needed.

Two strategies were adopted in this study for this purpose. The first determines the PL direction, giving the minimum sum of GDOPs estimated for all epochs. Thus, it aims to find the satellite-PL configuration providing the lowest total GDOP value for the considered time span. This strategy can be better explained in the following steps for clarity.

The first is to estimate a GDOP value for each direction covering the time span; second, to estimate the sums of the GDOP values based on the directions; third, to find the lowest total and the corresponding direction; and fourth, to use this direction for positioning of the PL.

The second strategy is based on the selection of the lowest GDOP among maximum GDOP values estimated for each direction.4 This is carried out in two stages: GDOP values are estimated for each direction for a 24-hour period and their maximum values are selected, and then the lowest GDOP value of the selected GDOPs is taken into consideration. Thus, the corresponding direction is determined.

Results

In order to validate the optimization approaches suggested in this study, one-day GPS satellite visibility data for Istanbul, Turkey (acquired above elevation angles of 23[degrees], which was set considering the topography of the study area and the visibility conditions), were used.

Visibility estimations were carried out at three-minute intervals for a 24-hour period (totaling 480 epochs), for which the GDOP values were estimated. Without a loss of generality, it is assumed that the elevation of the PL is zero (i.e., En=O). The azimuth angle of the PL is then used to estimate the GDOP values for all epochs.

As a result of this study, both strategies suggest close directions (277[degrees] and 278[degrees]) for the PL location. When the estimated GDOP values were analyzed, it was observed that the second strategy outperforms the first in terms of the maximum GDOP value.

The difference, however, is negligible. As a result, both methods can be successfully applied for the particular data set used in this study.

Results of the optimization strategies illustrate the variation in GDOP values. When the PL is placed at the center of the project area, it is estimated that the azimuth directions for the borders of the study area range from 240[degrees] to 60[degrees]. It was also found that the GDOP values for this range do not exceed 7.5. Various experiments conducted in the studies have shown that both methods may produce different results and suggest different directions for the positioning of a PL depending on the satellite geometry. The second method in particular, which is based on the analysis of the sum of the GDOP values, does not guarantee the minimum for GDOP maxima for all cases.

Conclusions

PLs provide extra and continuous signals to GPS users, which improves the positional accuracy in GPS applications. PLs transmit pseudorange and carrier phase signals at one or two GPS frequency bands (L1 or L2).

The use of PLs is particularly important for marine applications (e.g., hydrographic surveying, navigation in narrow waters and harbor entrances, indoor navigations and aircraft landing). Although they provide extra information and improve the quality of the results, their positioning is a crucial problem that needs further investigation. This study is an attempt to determine an optimal location for a PL augmented with GPS.

Two strategies based on different GDOP criteria were applied to data sets for Istanbul, Turkey. The first strategy is based on determining the direction of the PL by giving the minimum GDOP total for a 24-hour period. It searches a satellite-PL configuration providing the lowest total GDOP value for the considered time span. The second strategy discussed, on the other hand, searches the direction yielding minimum of the maximum GDOPs estimated for the 24- hour data.’ Both strategies were applied to the data set, and the results were thoroughly analyzed. It was observed that the methods produce comparable results and provide significant improvements over the case where no PL was employed.

Under the light of present results, it is recommended the results of the two techniques be analyzed together in order to benefit from their particular advantages.

(Second from Top) GDOP values for the safellites over elevation angles of 23[degrees] without a PL, with the pessimally positioned mobile PL (upper) and with the optimally positioned mobile PL (lower).

(Second From Bottom) Results of the optimization strategies for the data set.

(Bottom) Solutions of the optimization strategies using the minimum of maximum GDOPs (left) and the minimum of the sum of GDOPs (right).

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Dr. M. Halis Saka is an assistant professor, currently working in the Department of Geodetic and Photogrammetric Engineering at Cebze Institute of Technology in Turkey. He gives lectures on geodesy and global positioning system technology. His main research covers satellite positioning, navigation and geodesy.